Conditioning and solution of Hermitian (block) Toeplitz systems by means of preconditioned conjugate gradient methods

Stefano Serra-Capizzano

doi:10.1117/12.211409

7 June 1995 Conditioning and solution of Hermitian (block) Toeplitz systems by means of preconditioned conjugate gradient methods

Stefano Serra-Capizzano

Author Affiliations +

Proceedings Volume 2563, Advanced Signal Processing Algorithms; (1995) https://doi.org/10.1117/12.211409
Event: SPIE's 1995 International Symposium on Optical Science, Engineering, and Instrumentation, 1995, San Diego, CA, United States

Abstract

Let {A_n(f)} be a sequence of nested n X n Toeplitz matrices generated by a Lebesgue integrable real-function f defined on (-(pi) , (pi) ). In this paper, we first present some results about the spectral properties of A_n(f) (density, range, behavior of the extreme eigenvalues etc.), then we apply these results to the preconditioning problem. We analyze in detail the preconditioned conjugate gradient method, where the proposed poreconditioners A_n¹(g)A_n(f): we obtain new results about the range, the density and the extremal properties of their spectra. In particular we deal with the critical case where the matrices A_n(f) are asymptotically ill-conditioned, i.e., zero belongs to the convex hull of the essential range of f. We consider positive definite Toeplitz linear systems (f >= 0), nondefinite Toeplitz linear systems (f with nondefinite sign), with zeros of generic orders. Moreover, these analyses and techniques are partially extended to the block case, too.

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Stefano Serra-Capizzano "Conditioning and solution of Hermitian (block) Toeplitz systems by means of preconditioned conjugate gradient methods", Proc. SPIE 2563, Advanced Signal Processing Algorithms, (7 June 1995); https://doi.org/10.1117/12.211409

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